Finite Group Databasebeta

← #6802507

Subgroups — #6802507

Order 512.

1997 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3#32902C211yesmaximal, characteristic
4#32173C214yesmaximal, complemented
5#30739C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#28460C212yesmaximal, complemented
8#27748C212yesmaximal, complemented
9#25724C212yesmaximal, complemented
10#25718C211yesmaximal, characteristic, complemented
11#25367C212yesmaximal, complemented
12#24629C212yesmaximal, complemented
13#24215C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#495C2214yes—
22C2×#492C2212yescomplemented
23C2×#462C2212yescomplemented
24C2×#463C2214yes—
25C2×#459C2212yes—
26C2×#451C2212yes—
27C2×#448C2212yes—
28C2×#448C2212yes—
29C2×#450C2211yescharacteristic
30C2×#447C2212yes—
31C2×C4×SD16C2212yes—
32#2856C2211yescharacteristic
33#2850C2212yes—
34#2597—22no—
35#2571C2211yescharacteristic, complemented
36#2548C2211yescharacteristic, complemented
37#2547—22no—
38#2547—21no—
39#2547—21no—
40#2547C2214yes—
41#2547C2212yes—
42#2547C2212yes—
43#2513—21no—
44#2513C2212yescomplemented
45#2504—21no—
46#2447C2212yescomplemented
47#2444—22no—
48C2×#397—22no—
49#2366C2211yescharacteristic, complemented
50#2361—21no—
51#2357—21no—
52#2355C2211yescharacteristic, complemented
53C4×C22⋊Q8C2211yescharacteristic
54C4×#121C2212yes—
55C2×#391C2212yes—
56C2×#380—21no—
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2074C2214yescomplemented
60#2056C2214yescomplemented
61#2055C2214yescomplemented
62#2043C2212yescomplemented
63#2042C2214yes—
64#2016C2212yescomplemented
65#2015C2214yes—
66#2015C2214yescomplemented
67#2013C2214yescomplemented
68#2012C2214yes—
69#2012C2214yes—
70#1983C2214yes—
71#1981C2212yescomplemented
72#1968C2214yescomplemented
73#1958C2214yes—
74#1956C2212yescomplemented
75#1939C2212yescomplemented
76#1933C2214yescomplemented
77#1927C2214yescomplemented
78#1916C2214yescomplemented
79#1879C2214yes—
80#1877C2212yes—
81#1877C2214yescomplemented
82#1874C2212yes—
83#1872C2211yescharacteristic
84#1864C2212yes—
85#1853C2211yescharacteristic
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1786C2211yescharacteristic
91C8×C4⋊C4C2212yescomplemented
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×SD16C2312yes—
100C22×SD16C2314yes—
101C22×SD16C2314yes—
102C22×D8C2314yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124—22no—
110C2×#124—22no—
111C2×#124—48no—
112C2×#124C2314yes—
113C2×#124C2314yes—
114C2×#124C2312yes—
115C2×C22⋊Q8C2311yescharacteristic
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—44no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#495—48no—
139#492—44no—
140#463—48no—
141#462—44no—
142#459—44no—
143#451—44no—
144#450—24no—
145#448—28no—
146#448—44no—
147#447—44no—
148C4×SD16—44no—
149C2×#106C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×C22⋊C8C2311yescharacteristic
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2314yes—
168#400—22no—
169#397—22no—
170#397—22no—
171#397—24no—
172#397—24no—
173#397—24no—
174#397—24no—
175#397—22no—
176#397—22no—
177#394—22no—
178#391—22no—
179#391—22no—
180#391C2314yes—
181#391C2312yes—
182#391C2314yes—
183#391C2311yescharacteristic
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—21no—
193#389C2312yes—
194#389C2311yescharacteristic
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—21no—
201#387—22no—
202#386—22no—
203#386—22no—
204#386—22no—
205#386—22no—
206#385—21no—
207#385—22no—
208#384C2311yescharacteristic
209#382—21no—
210#382—21no—
211#382C2312yes—
212#381—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380—24no—
218#380—24no—
219#380C2311yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C25—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2411yescharacteristic
249C22×Q8C2414yes—
250C22×D4—24no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—42no—
258C22×D4C2412yes—
259C22×D4C2412yes—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—44no—
266C23×C4—44no—
267C23×C4—44no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273C2×Q16—44no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—28no—
279C2×SD16—42no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16—44no—
283C2×SD16—44no—
284C2×SD16—44no—
285C2×D8—28no—
286C2×D8—44no—
287C2×D8—44no—
288C22×C8C2414yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296#127—24no—
297#124—28no—
298#124—28no—
299#124—44no—
300#124—44no—
301#124—44no—
302#124—48no—
303#124—48no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306#121—24no—
307#121—24no—
308#121—44no—
309#121—82no—
310C4×Q8—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—84no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—44no—
366C2×C22⋊C4—48no—
367C2×C22⋊C4—48no—
368C2×C42—24no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—44no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381#106—42no—
382C4⋊C8—24no—
383C4⋊C8—44no—
384#103—28no—
385#103—42no—
386#103—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#102—24no—
391#102—24no—
392#102—42no—
393#102—44no—
394#102—44no—
395C22⋊C8—24no—
396C22⋊C8—22no—
397C22⋊C8#4114yes—
398C4×C8—24no—
399C4×C8—44no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—21no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—21no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418#95C2×D412yes—
419C24—24no—
420C24—22no—
421C24—22no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—21no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—88no—
443C2×Q8—88no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—48no—
455C2×D4—48no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4H(5,𝔽2)14yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—816no—
504C22×C4#13514yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4C2511yescharacteristic
517Q16—88no—
518SD16—48no—
519SD16—88no—
520SD16—88no—
521D8—88no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8C2×#4114yes—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—81no—
554C4⋊C4—82no—
555C4⋊C4—81no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C42—22no—
575C42—22no—
576C42—24no—
577C42—24no—
578C42—22no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—82no—
584C42H(5,𝔽2)14yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—24no—
590C23—48no—
591C23—48no—
592C23—48no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—84no—
607C23—84no—
608C23—84no—
609C23—84no—
610C23#55012yes—
611C23C2×#13512yes—
612C23#57712yes—
613C23C23×D411yescharacteristic
614Q8—42no—
615Q8—44no—
616Q8—44no—
617Q8—44no—
618Q8—42no—
619Q8—44no—
620Q8—88no—
621D4—24no—
622D4—24no—
623D4—48no—
624D4—42no—
625D4—44no—
626D4—42no—
627D4—44no—
628D4—88no—
629D4—84no—
630D4—162no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—88no—
682C2×C4—88no—
683C2×C4—164no—
684C2×C4#53512yes—
685C2×C4#53512yes—
686C2×C4#55012yes—
687C2×C4C2×#13512yes—
688C2×C4C2×#13512yes—
689C2×C4#57512yes—
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)12yes—
692C2×C4C2×H(5,𝔽2)12yes—
693C2×C4C2×H(5,𝔽2)11yescharacteristic
694C8—24no—
695C8—44no—
696C8—42no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—816no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22#339414yes—
719C22C2×#53511yescharacteristic
720C22C2×#55011yescharacteristic
721C22#348411yescharacteristic
722C22C22×#13511yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57711yescharacteristic
725C22#358611yescharacteristic
726C4—22no—
727C4—22no—
728C4—22no—
729C4—42no—
730C4—42no—
731C4—41no—
732C4—42no—
733C4—42no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4—84no—
740C4—81no—
741C4#321114yes—
742C4#351912yes—
743C4#353712yes—
744C2—24no—
745C2—84no—
746C2—162no—
747C2#3913211yescharacteristic
748C2C2×#321111yescharacteristic
749C2#6081311yescharacteristic
750C2C2×#339411yescharacteristic
751C2#6140311yescharacteristic
752C2#6156511yescharacteristic
753C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36587C2semidirect
2. C2×#2547C2semidirect
4. #32173C2semidirect
5. #30739C2semidirect
6. #29719C2semidirect
7. #28460C2semidirect
8. #27748C2semidirect
9. #25724C2semidirect
10. #25718C2semidirect
11. #25367C2semidirect
12. #24629C2semidirect
13. #24215C2semidirect
14. #24212C2semidirect
15. #22625C2semidirect
16. #22041C2semidirect
17. C4×#450C2semidirect
22. C2×#492C22semidirect
23. C2×#462C22semidirect
35. #2571C22semidirect
36. #2548C22semidirect
44. #2513C22semidirect
46. #2447C22semidirect
49. #2366C22semidirect
52. #2355C22semidirect
59. #2074C22semidirect
60. #2056C22semidirect
61. #2055C22semidirect
62. #2043C22semidirect
64. #2016C22semidirect
66. #2015C22semidirect
67. #2013C22semidirect
71. #1981C22semidirect
72. #1968C22semidirect
74. #1956C22semidirect
75. #1939C22semidirect
76. #1933C22semidirect
77. #1927C22semidirect
78. #1916C22semidirect
81. #1877C22semidirect
91. C8×C4⋊C4C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365872
C2×#25472
#329022
#321732
#307392
#297192
#284602
#277482
#257242
#257182
#253672
#246292
#242152
#242122
#226252
#220412
C4×#4502